Asymptotics for the Tukey depth process, with an application to a multivariate trimmed mean

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On the limiting distributions of multivariate depth-based rank sum statistics and related tests (2006, background): “Assumption (A2) in the theorem is satisfied by most depth functions such as the Mahalanobis, projection, simplicial and halfspace depth functions; see [9, 15, 25] for related discussions.”
Quantile functions for multivariate analysis: approaches and applications (2002, background): “Under similar conditions, and further assuming that F is centrally symmetric, asymptotic normality of ~L(F̂n) in the case of the halfspace depth function has been given by MASSÉ (2000).”
Multidimensional trimming based on projection depth (2006, background): “The numerator of equation (18) can be decomposed into three parts”
Depth weighted scatter estimators (2005, background): “Weighted or trimmed means based on the latter two depth functions were considered in Liu (1990), Dümbgen (1992) and Massé (2004).”
Location–Scale Depth (2004, background): “The asymptotic distribution theory of maximum depth estimators is a topic still under intense investigation; see He and Portnoy (1998), Bai and He (1999), and Massé (2002, 2004).”